A basis for free assosymmetric algebras (Q1921913)

The authors exhibit a natural basis for free assosymmetric algebras, i.e. algebras in which the associator \(\{x,y,z\}\) satisfies \(\{a,b,c\} = \{p(a),p(b),p(c)\}\) for all permutations \(p\) on \(a,b,c\). If \(F\) is a field of characteristic \(\neq 2\) and 3, \(S\) a set of generators, \(F[S]\) the free nonassociative algebra over \(F\) generated by \(S\), \(A\) the \(T\)-ideal in \(F[S]\) determined by the identities \(\{x,y,z\} - \{y,z,x\}\) and \(\{x,y,z\} - \{y,x,z\}\), then \(F[S]/A\) is a free assosymmetric algebra over \(F\). It is proved that the set \(P\) of standard polynomials over \(S\) forms a basis for this algebra. A standard polynomial is either a word of type \((..(x_1x_2.x_3)x_4)...)x_n\) (where \(x_i\) are generators) or an expression of type \[ a_{i_1} \Biggl(a_{i_2} \Bigl(\dots a_{i_m} \biggl[\dots \bigl[\{b_{j_1}, b_{j_2}, b_{j_3}\}, b_{j_4}\bigr], \dots, b_{j_n } \biggr] \dots \Bigr)\Biggr), \] where the arguments are generators with \(a_{i_1}\leq a_{i_2} \leq \cdots \leq a_{i_m}\), \(b_{j_1} \leq b_{j_2} \leq \cdots \leq b_{j_n}\) and \(\leq\) is assumed to be some arbitrary total order on \(S\). The multiplication of basis elements and the way in which arbitrary elements can be expressed relative to this basis are exhaustively described.